Analysis

Read a formula for what it permits

• Given a formula, the learner can identify every operation that restricts its domain, combine the resulting conditions, and express the answer in interval notation, distinguishing the domain a formula forces from the domain a situation permits.
• Given a linear, absolute-value, exponential or logarithmic inequality or equation, the learner can solve it, reverse the direction when multiplying or dividing by a negative quantity, unfold an absolute-value condition into the union or intersection it denotes, and reject candidates that fall outside the original expression's domain.
• Given a sequence defined by a formula or recursively, the learner can decide whether it converges, find the limit when it exists, produce an $N$ for a given $\epsilon$ when the limit is known, and establish existence by monotonicity and boundedness when no formula for the general term is available.
• The learner can evaluate and interpret $\ln x$ as the area under $1/t$, derive the product and power laws from that definition, differentiate and integrate expressions involving $\ln$, convert between bases, and identify where the laws fail or change a domain.

Compute trigonometric values and derive the identities

• Given an angle, the learner can locate it on the unit circle and give exact values for the special angles; given an identity, can derive it from the circle's equation or the angle-addition formulas rather than recalling it; and given a trigonometric equation, can find every solution in a stated interval and express the general solution.

Solve a differential equation

• Given a first-order ordinary differential equation, the learner can classify it as separable, linear or both, apply the corresponding method to obtain the general solution, use an initial condition to determine the constant, and verify the result by substitution.
• Given $y^{″}+p{y}^{\prime }+qy=0$, the learner can form the characteristic equation, classify its roots by the discriminant, write the general solution in the form that case requires, apply two initial conditions to determine both constants, and verify the result by substitution.
• Given ${\mathbf{x}}^{\prime }=A\mathbf{x}$ with constant $A$, the learner can compute the eigenvalues and eigenvectors of $A$, assemble the general solution from the eigenpairs, detect a defective repeated eigenvalue and supply the $t\mathbf{v}$ term it requires, determine the constants from an initial vector, and read the long-run behaviour from the eigenvalues alone.
• Given an initial value problem, the learner can carry out Euler and improved Euler steps by hand, evaluating each slope at the point the formula names and tabulating the result against an exact solution where one exists.
• Given a numerical result or a problem to integrate, the learner can predict how the error responds to halving the step from the method's order, compute the stability limit and recognise when it binds rather than accuracy, select a method and step size for a stated requirement, and say what the computed output does and does not establish.